arxiv: 2605.15095 · v1 · submitted 2026-05-14 · 🧮 math.GT · math.SG

Recognition: 2 theorem links

· Lean Theorem

Mazur manifolds and symplectic structures

Alberto Cavallo

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Pith reviewed 2026-05-15 02:55 UTC · model grok-4.3

classification 🧮 math.GT math.SG

keywords Mazur manifoldssymplectic structuresHeegaard Floer homologyBrieskorn spherescobordism mapsexotic 4-manifoldsdefinite intersection forms

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The pith

Heegaard Floer cobordism maps obstruct symplectic structures on specific Akbulut-Kirby Mazur manifolds bounded by Brieskorn spheres.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that three particular Akbulut-Kirby Mazur manifolds cannot carry symplectic structures when their boundary is one of the Brieskorn spheres Σ(2,3,13), Σ(2,5,7), or Σ(3,4,5). It reaches this by computing the Heegaard Floer homology cobordism maps induced by the 2-handles in each manifold's construction and showing that these maps violate the algebraic conditions required of any symplectic filling. A reader would care because the result supplies explicit topological 4-manifolds that are provably non-symplectic despite having simple handle decompositions. The same maps further imply the existence of exotic pairs of simply connected 4-manifolds that share the same definite intersection form and the same boundary Y. The argument works directly from the Floer data of the given handles and spheres without invoking extra structure on possible fillings.

Core claim

The Akbulut-Kirby Mazur manifolds whose boundary is a Brieskorn sphere Y among Σ(2,3,13), Σ(2,5,7) and Σ(3,4,5) do not admit symplectic structures. This follows from the fact that the Heegaard Floer homology cobordism maps associated to the 2-handles in their handle decompositions fail to satisfy the properties that any symplectic filling would impose on the Floer groups. The same obstruction yields exotic pairs of simply connected 4-manifolds with definite intersection forms that are bounded by each such Y.

What carries the argument

The Heegaard Floer homology cobordism maps induced by the 2-handles of the Mazur manifold.

If this is right

The three listed Akbulut-Kirby Mazur manifolds admit no symplectic structure.
For each such boundary Y there exist exotic pairs of simply connected 4-manifolds carrying the same definite intersection form.
The obstruction arises precisely because the relevant cobordism maps are nonzero in a way forbidden by symplectic geometry.
The method applies uniformly to the three chosen Brieskorn spheres using only their Floer data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same cobordism-map technique may rule out symplectic structures on other Mazur manifolds or handlebodies with known Floer data.
These examples suggest that definite intersection forms on 4-manifolds bounded by Brieskorn spheres can be realized by non-symplectic manifolds in multiple ways.
Contact invariants on the boundary spheres might be related to the same maps, though the paper does not compute them.

Load-bearing premise

The Heegaard Floer cobordism maps for the specific 2-handles and chosen Brieskorn spheres detect the absence of a symplectic structure without additional assumptions on the filling or on the contact structure induced on the boundary.

What would settle it

An explicit symplectic filling of one of the listed Mazur manifolds whose induced map on Heegaard Floer homology matches the computed cobordism map would falsify the obstruction for that manifold.

Figures

Figures reproduced from arXiv: 2605.15095 by Alberto Cavallo.

**Figure 1.** Figure 1: The standard graph of Σ(2, 3, 13) (left), Σ(2, 5, 7) (middle) and Σ(3, 4, 5) (right). It is a result of Mark and Tosun [19, Theorem 1.8] that the Akbulut-Kirby Mazur manifold W±(2), whose boundary is ±Σ(2, 3, 13), does not carry a Stein structure. In this paper we focus on Σ(2, 3, 13), Σ(2, 5, 7) and Σ(3, 4, 5), whose plumbing graph is drawn in [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

**Figure 2.** Figure 2: The Mazur manifold W±(ℓ, k), and the knot γ± on its boundary. Note that the diffeomorphism type of W−(ℓ, k) only depend on ℓ + k. We recall the following result which is deduced from [4, Proposition 1]. We write that X1 is (orientation-preserving) diffeomorphic to X2 by X1 ∼= X2. Theorem 1.2 (Akbulut-Kirby) The Mazur manifold W ±(ℓ, k) satisfies the following properties: (1) W±(ℓ, k) ∼= W±(ℓ + 1, k − 1) fo… view at source ↗

**Figure 3.** Figure 3: Stein structures on W−(0, 1 − s) (left) and W+(1, −4 − s) (right). It is easy to check that the Legendrian knots where we attach the Stein 2-handles have tb-number equal to 2 − s and −3 − s, where s is the number of stabilizations. Ozsváth and Szabó [21] defined the cobordism map F ◦ W,u : HF◦ (S 3 ) → HF◦ (Y := ∂W, u|Y ) induced by any 4-manifold W, equipped with a Spinc -structure u; in particular, when … view at source ↗

**Figure 4.** Figure 4: The self-diffeomorphism F of Y −(3) mapping K3 to γ−. There are four diffeomorphisms fA, fB, fC and Φ2,1 involved: we have that fA(γ ′ +) = K3, fB(γ ′ +) = γ ′ +, fC (γ+) = γ ′ + while Φ2,1(γ−) = γ+ is the identification Y −(2, 1) ∼= −Y +(0, 0) established in Property 2) of Theorem 1.2. The diffeomorphism we want is F = Φ−1 2,1 ◦ f −1 C ◦ fB ◦ f −1 A . We start from the case of m = 3, corresponding to Σ(2… view at source ↗

**Figure 5.** Figure 5: The self-diffeomorphism F of Y −(4) mapping K4 to γ−. There are three diffeomorphisms fD, fE and Φ2,2 involved: we have that fD(K4) = γ ′ +, fE(γ+) = γ ′ + while Φ2,2(γ−) = γ+ is the identification Y −(2, 2) ∼= −Y +(0, −1) established in Property 2) of Theorem 1.2. The diffeomorphism we want is F = Φ−1 2,2 ◦ f −1 E ◦ fD [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: The Stein domain X(2) whose boundary is Σ(2, 3, 13). When m = 2 we recall that from the proof of Proposition 3.1 one has that FbW−(2)(1) = θ ∈ HFd(Y −(2)) is either [V0] or [V1] + [V0] + [V−1], where {[V1], [V0], [V−1]} is the canonical basis of [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

read the original abstract

We use the Heegaard Floer homology cobordism maps to obstruct the existence of a symplectic structure on the Akbulut-Kirby Mazur manifolds whose boundary is a Brieskorn sphere $Y$ among $\Sigma(2,3,13),$ $\Sigma(2,5,7)$ and $\Sigma(3,4,5)$. Furthermore, we describe how our results imply the existence of exotic pairs of simply connected 4-manifolds, with definite intersection form, whose boundary is $Y$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Cavallo computes HF cobordism maps for three specific Akbulut-Kirby Mazur manifolds to obstruct symplectic structures and produce new exotic definite 4-manifolds.

read the letter

The main point is that Cavallo uses Heegaard Floer homology cobordism maps to obstruct symplectic structures on three Akbulut-Kirby Mazur manifolds with Brieskorn sphere boundaries, specifically Σ(2,3,13), Σ(2,5,7), and Σ(3,4,5). This also leads to exotic pairs of simply connected 4-manifolds with definite intersection forms. The work applies standard HF techniques to these new examples and produces concrete obstructions. It does not introduce a general framework but instead checks specific cases that had not been looked at before. That is useful because it adds to the list of known exotic 4-manifolds in this class. The arguments rely on established properties of the cobordism maps, so there is no circularity or invented assumptions. The computations of the maps for the 2-handles seem to be the core, and they are said to show incompatibility with symplectic fillings through the boundary contact structures or module structures. This matches what the field accepts for such results. One soft spot is that without the full details of the map calculations in front of me, I cannot double-check the ranks or non-vanishing claims for these particular spheres. A referee should verify those steps to make sure the obstruction is as strong as stated. The step from obstruction to exotic pairs also needs clear explanation of how the manifolds differ. Overall, this paper is aimed at people working on 4-manifold topology and symplectic geometry. It gives new examples using reliable tools, so it is worth a serious referee's time. I recommend sending it to peer review.

Referee Report

0 major / 2 minor

Summary. The paper claims to use Heegaard Floer homology cobordism maps induced by 2-handle attachments to obstruct the existence of symplectic structures on the Akbulut-Kirby Mazur manifolds whose boundaries are the Brieskorn spheres Σ(2,3,13), Σ(2,5,7), and Σ(3,4,5). It further derives from these obstructions the existence of exotic pairs of simply connected 4-manifolds with definite intersection forms that share these boundaries.

Significance. If the cobordism map computations are correct, the results supply explicit new examples of non-symplectic Mazur manifolds and rare exotic pairs of definite 4-manifolds with the same boundary, strengthening the literature on symplectic fillings of Brieskorn spheres via standard HF tools without new axioms or fitted parameters.

minor comments (2)

[Abstract] The abstract refers to 'Akbulut-Kirby Mazur manifolds' without naming the precise handle attachments or diagrams; a sentence or reference to the relevant figures in the introduction would aid readers unfamiliar with the specific presentations.
[Introduction] Notation for the Brieskorn spheres is consistent, but the manuscript should explicitly recall the grading or module structure of HF^+(Y) for each Y in a preliminary section to make the obstruction arguments self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending acceptance. The referee's summary accurately describes our use of Heegaard Floer cobordism maps to obstruct symplectic structures on the indicated Mazur manifolds and the consequent construction of exotic definite pairs.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper computes explicit Heegaard Floer cobordism maps induced by 2-handle attachments in the Mazur manifold presentations for the listed Brieskorn spheres and uses their non-vanishing or rank properties to obstruct symplectic fillings. These maps rely on standard, externally established properties of HF homology from prior literature (Ozsváth-Szabó and related works) rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The obstruction follows directly from the module structure and grading already known for these spheres, with no reduction of the target claim to the paper's own inputs by construction. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard axioms of Heegaard Floer homology for 3-manifolds and cobordisms, together with the topological description of the Akbulut-Kirby Mazur manifolds; no new free parameters or invented entities are introduced.

axioms (1)

standard math Heegaard Floer homology cobordism maps are well-defined and functorial for the relevant 4-dimensional cobordisms
Invoked throughout the argument as the source of the obstruction

pith-pipeline@v0.9.0 · 5365 in / 1342 out tokens · 30277 ms · 2026-05-15T02:55:29.613335+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We use the Heegaard Floer homology cobordism maps to obstruct the existence of a symplectic structure on the Akbulut-Kirby Mazur manifolds
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the cobordism map bF_W-(m) : dHF(S^3) → dHF_0(Y-(m)) sends the generator to [V0]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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